Calculus Examples

$\int_{0}^{2} x \sqrt{4 - x^{2}} d x$

Step 1

Let $u = 4 - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 - x^{2}$ .

$\frac{d}{d x} [4 - x^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $4 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$

Step 1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 1.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 4 - x^{2}$ .

$u_{lower} = 4 - 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 4 - 0$

Step 1.3.1.2

Multiply $- 1$ by $0$ .

$u_{lower} = 4 + 0$

Step 1.3.2

Add $4$ and $0$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = 4 - x^{2}$ .

$u_{upper} = 4 - 2^{2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $2$ to the power of $2$ .

$u_{upper} = 4 - 1 \cdot 4$

Step 1.5.1.2

Multiply $- 1$ by $4$ .

$u_{upper} = 4 - 4$

Step 1.5.2

Subtract $4$ from $4$ .

$u_{upper} = 0$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4$

$u_{upper} = 0$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{4}^{0} \sqrt{u} \frac{1}{- 2} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{4}^{0} \sqrt{u} (- \frac{1}{2}) d u$

Step 2.2

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

$\int_{4}^{0} - \frac{\sqrt{u}}{2} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{4}^{0} \frac{\sqrt{u}}{2} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int_{4}^{0} \sqrt{u} d u)$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$- \frac{1}{2} \int_{4}^{0} u^{\frac{1}{2}} d u$

Step 6

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$- \frac{1}{2} \frac{2}{3} u^{\frac{3}{2}}]_{4}^{0}$

Step 7

Simplify the expression.

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Step 7.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $0$ and at $4$ .

$- \frac{1}{2} ((\frac{2}{3} \cdot 0^{\frac{3}{2}}) - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 7.2

Simplify.

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Step 7.2.1

Rewrite $0$ as $0^{2}$ .

$- \frac{1}{2} (\frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 7.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{2} (\frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 7.2.3

Cancel the common factor of $2$ .

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Step 7.2.3.1

Cancel the common factor.

$- \frac{1}{2} (\frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 7.2.3.2

Rewrite the expression.

$- \frac{1}{2} (\frac{2}{3} \cdot 0^{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 7.2.4

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{2} (\frac{2}{3} \cdot 0 - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 7.3

Multiply $\frac{2}{3}$ by $0$ .

$- \frac{1}{2} (0 - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 7.4

Simplify.

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Step 7.4.1

Rewrite $4$ as $2^{2}$ .

$- \frac{1}{2} (0 - \frac{2}{3} \cdot {(2^{2})}^{\frac{3}{2}})$

Step 7.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{2} (0 - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})})$

Step 7.4.3

Cancel the common factor of $2$ .

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Step 7.4.3.1

Cancel the common factor.

$- \frac{1}{2} (0 - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})})$

Step 7.4.3.2

Rewrite the expression.

$- \frac{1}{2} (0 - \frac{2}{3} \cdot 2^{3})$

Step 7.4.4

Raise $2$ to the power of $3$ .

$- \frac{1}{2} (0 - \frac{2}{3} \cdot 8)$

Step 7.5

Simplify.

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Step 7.5.1

Multiply $8$ by $- 1$ .

$- \frac{1}{2} (0 - 8 (\frac{2}{3}))$

Step 7.5.2

Combine $- 8$ and $\frac{2}{3}$ .

$- \frac{1}{2} (0 + \frac{- 8 \cdot 2}{3})$

Step 7.5.3

Multiply $- 8$ by $2$ .

$- \frac{1}{2} (0 + \frac{- 16}{3})$

Step 7.5.4

Move the negative in front of the fraction.

$- \frac{1}{2} (0 - \frac{16}{3})$

Step 7.6

Subtract $\frac{16}{3}$ from $0$ .

$- \frac{1}{2} (- \frac{16}{3})$

Step 7.7

Simplify.

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Step 7.7.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) \frac{16}{3}$

Step 7.7.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} \cdot \frac{16}{3}$

Step 7.7.3

Multiply $\frac{1}{2}$ by $\frac{16}{3}$ .

$\frac{16}{2 \cdot 3}$

Step 7.7.4

Multiply $2$ by $3$ .

$\frac{16}{6}$

Step 7.7.5

Cancel the common factor of $16$ and $6$ .

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Step 7.7.5.1

Factor $2$ out of $16$ .

$\frac{2 (8)}{6}$

Step 7.7.5.2

Cancel the common factors.

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Step 7.7.5.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 8}{2 \cdot 3}$

Step 7.7.5.2.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 \cdot 3}$

Step 7.7.5.2.3

Rewrite the expression.

$\frac{8}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{3}$

Decimal Form:

$2.\overline{6}$

Mixed Number Form:

$2 \frac{2}{3}$